Problem: Simplify and expand the following expression: $ \dfrac{1}{n + 1}+ \dfrac{3}{5n + 20}- \dfrac{2}{n^2 + 5n + 4} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{3}{5n + 20} = \dfrac{3}{5(n + 4)}$ We can factor the quadratic in the third term: $ \dfrac{2}{n^2 + 5n + 4} = \dfrac{2}{(n + 1)(n + 4)}$ Now we have: $ \dfrac{1}{n + 1}+ \dfrac{3}{5(n + 4)}- \dfrac{2}{(n + 1)(n + 4)} $ The least common multiple of the denominators is: $ (n + 1)(n + 4)$ In order to get the first term over $(n + 1)(n + 4)$ , multiply by $\dfrac{5(n + 4)}{5(n + 4)}$ $ \dfrac{1}{n + 1} \times \dfrac{5(n + 4)}{5(n + 4)} = \dfrac{5(n + 4)}{(n + 1)(n + 4)} $ In order to get the second term over $(n + 1)(n + 4)$ , multiply by $\dfrac{n + 1}{n + 1}$ $ \dfrac{3}{5(n + 4)} \times \dfrac{n + 1}{n + 1} = \dfrac{3(n + 1)}{(n + 1)(n + 4)} $ In order to get the third term over $(n + 1)(n + 4)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{2}{(n + 1)(n + 4)} \times \dfrac{5}{5} = \dfrac{10}{(n + 1)(n + 4)} $ Now we have: $ \dfrac{5(n + 4)}{(n + 1)(n + 4)} + \dfrac{3(n + 1)}{(n + 1)(n + 4)} - \dfrac{10}{(n + 1)(n + 4)} $ $ = \dfrac{ 5(n + 4) + 3(n + 1) - 10} {(n + 1)(n + 4)} $ Expand: $ = \dfrac{5n + 20 + 3n + 3 - 10}{5n^2 + 25n + 20} $ $ = \dfrac{8n + 13}{5n^2 + 25n + 20}$